# Step-by-step Solution

## Integral of (2x-1)/((x-1)(x-2)*(2x-3))

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### Videos

$\ln\left|x-1\right|+3\ln\left|x-2\right|-4\ln\left|2x-3\right|+C_0$

## Step-by-step explanation

Problem to solve:

$\int\frac{2x-1}{\left(x-1\right)\left(x-2\right)\left(2x-3\right)}dx$
1

Rewrite the fraction $\frac{2x-1}{\left(x-1\right)\left(x-2\right)\left(2x-3\right)}$ in $3$ simpler fractions using partial fraction decomposition

$\frac{2x-1}{\left(x-1\right)\left(x-2\right)\left(2x-3\right)}=\frac{A}{x-1}+\frac{B}{x-2}+\frac{C}{2x-3}$
2

Find the values of the unknown coefficients. The first step is to multiply both sides of the equation by $\left(x-1\right)\left(x-2\right)\left(2x-3\right)$

$2x-1=\left(x-1\right)\left(x-2\right)\left(2x-3\right)\left(\frac{A}{x-1}+\frac{B}{x-2}+\frac{C}{2x-3}\right)$

$\ln\left|x-1\right|+3\ln\left|x-2\right|-4\ln\left|2x-3\right|+C_0$
$\int\frac{2x-1}{\left(x-1\right)\left(x-2\right)\left(2x-3\right)}dx$